Explicit matrix coefficients and test vectors for discrete series representations

TL;DR

This paper introduces explicit matrix coefficients and test vectors for discrete series representations of GL(n) over non-archimedean fields, providing new tools for understanding their distinction and explicit formulas.

Contribution

It defines functions analogous to zonal spherical functions for discrete series, proves their existence in level 0, and offers explicit test vectors and coefficients for distinguished representations.

Findings

Established existence of special functions for discrete series at level 0

Provided explicit coefficients for unramified principal series

Presented a local proof of Matringe's criterion for distinction

Abstract

For the discrete series representations of ${\rm GL}(n)$ over a non-archimedean local field $F$, we define a notion of functions similar to "zonal spherical functions" for unramified principal series. We prove the existence of such functions in the level $0$ case. As for unramified principal series, they give rise to explicit coefficients. We deduce a local proof of Matringe's criterion of distinction of discrete series, in the level $0$ case, for the Galois symmetric space ${\rm GL}(n,F)/{\rm GL}(n,F_0 )$, for any unramified quadratic extension $F/F_0$. We also exhibit explicit test vectors when these representations are distinguished.